MCQ
If $A = \left[ {\begin{array}{*{20}{c}}\lambda &1\\{ - 1}&{ - \lambda }\end{array}} \right]$, then for what value of $\lambda ,\,{A^2} = O$
- A$0$
- ✓$ \pm {\rm{ }}1$
- C$-1$
- D$1$
✓
Answer
Correct option: B.
$ \pm {\rm{ }}1$
b
(b) ${A^2} = A\,.\,A = \left[ {\begin{array}{*{20}{c}}\lambda &1\\{ - 1}&{ - \lambda }\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}\lambda &1\\{ - 1}&{ - \lambda }\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{\lambda ^2} - 1}&0\\0&{ - 1 + {\lambda ^2}}\end{array}} \right] = 0$
(b) ${A^2} = A\,.\,A = \left[ {\begin{array}{*{20}{c}}\lambda &1\\{ - 1}&{ - \lambda }\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}\lambda &1\\{ - 1}&{ - \lambda }\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{\lambda ^2} - 1}&0\\0&{ - 1 + {\lambda ^2}}\end{array}} \right] = 0$
(As given)
==> ${\lambda ^2} - 1 = 0 \Rightarrow \lambda = \pm 1$.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If $m$ is the $A.M$ of two distinct real numbers $ l$ and $n (l,n>1) $ and $G_1, G_2$ and $G_3$ are three geometric means between $l$ and $n$ then $G_1^4 + 2G_2^4 + G_3^4$ equals :
→If $p = 7i - 2j + 3k$ and $q = 3i + j + 5k,$ then the magnitude of $p - 2q$ is
→$\mathop {\lim }\limits_{x \to {0^ + }} {\left( {\sec \sqrt x } \right)^{\frac{{10}}{x}}}$ is equal to
→The number of shortest paths from point $A\ to\ D$ (as shown in figure)
→If all the words with or without meaning made using all the letters of the word $" \text{NAGPUR}"$ are arranged as in a dictionary, then the word at $315^{th}$ position in this arrangement is :
→Three vertices are chosen randomly from the seven vertices of a regular $7$ -sided polygon. The probability that they form the vertices of an isosceles triangle is
→Let $a_1, a_2 , a_3,.....$ be an $A.P$, such that $\frac{{{a_1} + {a_2} + .... + {a_p}}}{{{a_1} + {a_2} + {a_3} + ..... + {a_q}}} = \frac{{{p^3}}}{{{q^3}}};p \ne q$. Then $\frac{{{a_6}}}{{{a_{21}}}}$ is equal to
→Let $\mathrm{ABC}$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $\mathrm{ABC}$ and the same process is repeated infinitely many times. If $\mathrm{P}$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then:
→Let $a, b, c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax + 2ay + c = 0$ and $5bx + 2by + d =0$ lies in the fourth quadrant and is equidistant from the two axes then
→The sum of all the elements of the set $\{\alpha \in\{1,2, \ldots, 100\}: \operatorname{HCF}(\alpha, 24)=1\}$ is
→